Hello Aondiar

Welcome to Math Help Forum!I'm not sure that I fully understand the details of your description of the problem, but if you take a look at the attached diagram, you can see whether it's what you want.

The square has been rotated through an angle anticlockwise, where . For instance, , giving . The square has then been reduced in size so that its vertices lie on the perimeter of the original square, to produce .

Then, as shown in the diagram, has coordinates

The coordinates of the remaining vertices are then fairly obvious.

Let me know if the diagram is not what you meant.

Grandad